RiskMetrics™ —Technical Document

68 Chapter 4. Statistical and probability foundations
where E[ ] denotes the mathematical expectation. Two additional measures that we will make reference
to within this document are known as skewness and kurtosis. Skewness characterizes the
asymmetry of a distribution around its mean. The expression for skewness is given by
[4.36]
s 3 = E ( r t
– µ ) 3 (skewness)
For the normal distribution skewness is zero. In practice, it is more convenient to work with the
skewness coefficient which is defined as
[4.37]
γ =
E ( r t
– µ ) 3
-------------------------------- (skewness coefficient)
σ 3
Kurtosis measures the relative peakedness or flatness of a given distribution. The expression for
kurtosis is given by
[4.38]
s 4 = E ( r t
– µ ) 4 (kurtosis)
As in the case of skewness, in practice, researchers frequently work with the kurtosis coefficient
defined as
[4.39]
κ=
E ( r t
– µ ) 4
-------------------------------- (kurtosis coefficent)
σ 4
For the normal distribution, kurtosis is 3. This fact leads to the definition of excess kurtosis which
is defined as kurtosis minus 3.
4.5.2.2 Using percentiles to measure market risk
Market risk is often measured in terms of a percentile (also referred to as quantile) of a portfolio’s
return distribution. The attractiveness of working with a percentile rather than say, the variance
of a distribution, is that a percentile corresponds to both a magnitude (e.g., the dollar amount
at risk) and an exact probability (e.g., the probability that the magnitude will not be exceeded).
The pth percentile of a distribution of returns is defined as the value that exceeds p percent of the
returns. Mathematically, the pth percentile (denoted by α) of a continuous probability distribution,
is given by the following formula
[4.40]
α
∫
p = f ( r) dr
–∞
where f (r) represents the PDF (e.g., Eq. [4.34])
So for example, the 5th percentile is the value (point on the distribution curve) such that 95 percent
of the observations lie above it (see Chart 4.18).
When we speak of percentiles they are often of the percentiles of a standardized distribution,
which is simply a distribution of mean-centered variables scaled by their standard deviation. For
2
example, suppose the log price change r t is normally distributed with mean µ t and variance σ t
.
The standardized return r˜t is defined as
RiskMetrics —Technical Document
Fourth Edition